Our observer design is most conveniently carried out using the state space form of the neural mass models. However, some of the neural mass models of interest presented in Section 2.3 were in block diagram form (e.g. Figure 2.3, 2.5 and 2.7) in the given references [74;138;155]. In [74] and [155], state space forms were provided but not in the convenient state coordinates where the techniques we use for proving convergence
2. A class of neural mass models
of estimates can be applied. Therefore, we illustrate how this can be done for the model by Wendling et al., whose detailed block diagram can be found in Figure2.3. The other models considered are special cases of the model by Wendling et al. and hence, can easily be obtained from the derivation below.
We will show that all neural mass models from Section 2.3 can be written in the following state space form:
˙x = Ax + G(p?)γ(Hx) + σ(u, Cx, p?), (2.8) and the output of the model is:
y = Cx, (2.9)
where the state vector is x∈ Rnx, input is u∈ R, output/EEG measurement is y ∈ R, parameter vector is p? ∈ Rnp, G : Rnp → Rnx×m, nonlinearity γ = (γ1, . . . , γm) with γi : R→ R for i ∈ {1, . . . , m} and nonlinearity σ = (σ1, . . . , σn) with σi : R×R×Rm → R for i∈ {1, . . . , n}. The number of states nx, number of parameters np and number of scalar nonlinear functions m differs for each model. These are defined in Sections 2.4.2-2.4.4.
2.4.1 Physiological interpretation
Physiologically, the first term in (2.8) implements the postsynaptic potential (PSP) kernels from (2.1), (2.2) and (2.3). This is effectively a convolution of the pre-synaptic firing rates arriving from other populations with the appropriate PSP response func-tions. These firing rates are modelled in the second and third term in (2.8) that incorporates the sigmoid firing rate function of the depolarisation of contributing pop-ulations. The second term, G(p)γ(Hx), reflects the influence of all states except the membrane potential of pyramidal population Cx. While the third term, σ(u, Cx, p), reflects the influence of the mean membrane potential of the pyramidal cells Cx and the exogenous input u. Neurobiologically, G(p)γ(Hx) + σ(u, Cx, p) correspond to the effects of intrinsic and extrinsic connections. In other words, when coupling different neural mass models, one has to consider the (intrinsic) influence of populations within a neural mass model and (extrinsic) contributions from other neural mass models. The extrinsic contributions are usually mediated through pyramidal cell populations.
In the following sections, we present the state space form for each model for ease of observer design. Detailed derivations are first shown for the model by Wendling et
2. A class of neural mass models
al., as it is the most complex model and then the subtle differences in derivations are described for the other models.
2.4.2 State space form for the model by Wendling et al.
We write the Wendling et al. model in state space form by introducing the state vari-ables xi1for i∈ {1, . . . , 7} as the membrane potential contribution from one population to another and xi2 for i ∈ {1, . . . , 7} as its derivative. The states xi1 are introduced at the outputs of all the impulse responses he, hi and hg blocks as shown in Figure 2.3. Recalling that the Laplace transform of the impulse responses he, hi and hg (as described by (2.1), (2.2) and (2.3)) are second-order transfer functions, by performing the inverse Laplace transform, each transfer function is represented by a second-order ordinary differential equation (ODE). We show this transformation for he from (2.1) as an example. Let the input to the he block be ¯u and output be ¯y. We denote the Laplace transform of signal v as L(v). Hence, the Laplace transform of he with zero initial conditions is:
L(he(t)) =L(θAat exp(−at)) = θAa
(s + a)2. (2.10)
Recalling thatL(he) = L(¯u)L(¯y), we obtain:
L(¯y)s2+ 2aL(¯y)s + a2L(¯y) = θAaL(¯u). (2.11)
By taking the inverse Laplace transform, we obtain a second-order ODE as follows:
¨¯
y + 2a ˙¯y + a2y = θ¯ Aa¯u. (2.12) The xi2 states are defined as xi2 = ˙xi1 for i ∈ {1, . . . , 7} to rewrite the second-order ODEs as two first-order ODEs for each impulse response block.
We illustrate this for the he(t) block in the fast inhibitory population, then the output of that block is ¯y = x51 and the input is ¯u = C3S(x11− x21− x31). Taking x52= ˙x51, (2.12) can be written as two first-order ODE as follows:
˙x51 = x52
˙x52 = −2ax52− a2x51+ θAaC3S(x11− x21− x31).
Hence, each impulse response he, hi and hg will each introduce a first-order ODE
2. A class of neural mass models
in the following general state space form by taking xi= (xi1, xi2) for i∈ {1, . . . , 7}:
˙xi = Aixi+ (0, ϑiS(µi) + ϕi) , (2.13) where µi is the input to the respective sigmoid functions,
Ai =
"
0 1
−ki1ki2 −(ki1+ ki2)
#
, for i = {1, . . . , 7} with k11 = k41 = k51 = k61 = a, k12 = k42 = k52 = k62 = a, k21 = k71 = b, k22 = k72 = b, k31 = g and k32 = g. ϑi
and ϕi are defined as such ϑ1 = θAaC2, ϑ2 = θBbC4, ϑ3 = θGgC7, ϑ7 = θBbC6, ϑ4 = ϑ5 = ϑ6 = 0 and ϕ1 = θAau, ϕ2 = ϕ3 = ϕ7 = 0, ϕ4 = θAaC1S(y), ϕ5 = θAaC3S(y), ϕ6 = θAaC5S(y). Constants a, b and g are strictly positive. S is a sigmoid function described by (2.4). All constants discussed in this section are summarised in B.12.
The subsystems defined in (2.13) are put together to be written compactly in state space form (2.8)-(2.9) for ease of observer design.
We take the state vector in (2.8) and (2.9) to be x = (x1, . . . , x7) where xi for i ={1, . . . , 7} satisfy (2.13). The states x1, x2 and x3 capture the membrane potential contribution and its derivative of the excitatory, slow and fast inhibitory populations to the pyramidal neurons, respectively. The states x4, x5 and x6 capture the membrane potential contribution and its derivative of the pyramidal neurons to the excitatory, slow and fast inhibitory populations, respectively. The output is y = x11− x21− x31. The specific matrices in (2.8) and (2.9) are denoted as:
• The parameter vector is p? = (θA, θB, θG),
• The matrix A = diag(A1, . . . , A7),
• γ = (S, S, S), where S is defined in (2.4),
• σ = (0, θAau, 0, 0, 0, 0, 0, θAaC1S(y), 0, θAaC3S(y), 0, θAaC5S(y), 0, 0), where S is described by (2.4),
• C = [ 1 0 −1 0 −1 0 0 0 0 0 0 0 0 0 ],
2. A class of neural mass models
2.4.3 State space form for the model by Jansen and Rit
We write the model in state space form by taking the state vector in (2.8) to be x = (x1, x2, x4, x5), where xi for i = {1, 2, 4, 5} satisfy (2.13). States x1 and x2 are the membrane potential contribution and its derivative of the excitatory and inhibitory populations to the pyramidal neurons, respectively. States x4 and x5 capture the membrane potential contribution and its derivative of the pyramidal neurons to the excitatory and inhibitory populations, respectively. The output is y = x11− x21. The specific matrices in (2.8) and (2.9) are denoted as:
• The parameter vector is p? = (θA, θB, C1, C2, C3, C4),
• A = diag(A1, A2, A4, A5),
• γ = (S, S) where S is defined in (2.4),
• σ = (0, θAau, 0, 0, 0, θAaC1S(y), 0, θAaC3S(y)),
• C = [ 1 0 −1 0 0 0 0 0 ],
2. A class of neural mass models
2.4.4 State space form for the model by Stam et al.
The model is written in state space form by taking the state vector in (2.8) as x = (x1, x2, x5) where xi for i = {1, 2, 5} satisfy (2.13). State x1 = (x11, x12) represents the mean membrane potential of the excitatory population’s activity to itself and its derivative, respectively. States x2 = (x21, x22) and x5 = (x51, x52) represent the mean membrane potential and its derivative of the inhibitory population to the excitatory population and vice versa, respectively. The output is y = x11− x21.
As mentioned in Section2.3.3, the PSP kernels heand hi differ from the ones in the models by Wendling and Jansen et. al.. Nevertheless, rewriting (2.5)-(2.7) into state space form does not differ from the derivation presented in Section2.4. By following the procedure described in Section 2.4.2, i.e. by taking Laplace transformations of (2.5) and (2.6) and taking the inverse Laplace transform, the kernels can be written as second-order ODEs. They can then be rewritten as two first-order ODEs by introducing extra state variables, xi2 for i∈ {1, 2, 5}, in a similar fashion as in Section 2.4.2. The specific matrices in (2.8) are denoted as:
• The parameter vector is p? = (C3, C4),
2. A class of neural mass models
• H = [ 0 0 0 0 1 0 ].